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Constancy of generalized Hodge–Tate weights of a local system

Part of: Arithmetic problems. Diophantine geometry

Published online by Cambridge University Press:  06 November 2018

Koji Shimizu
Koji Shimizu*
Affiliation:
Department of Mathematics, University of California, Berkeley, Evans Hall, Berkeley, CA 94720, USA email shimizu@math.berkeley.edu
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Abstract

Sen attached to each $p$-adic Galois representation of a $p$-adic field a multiset of numbers called generalized Hodge–Tate weights. In this paper, we discuss a rigidity of these numbers in a geometric family. More precisely, we consider a $p$-adic local system on a rigid analytic variety over a $p$-adic field and show that the multiset of generalized Hodge–Tate weights of the local system is constant. The proof uses the $p$-adic Riemann–Hilbert correspondence by Liu and Zhu, a Sen–Fontaine decompletion theory in the relative setting, and the theory of formal connections. We also discuss basic properties of Hodge–Tate sheaves on a rigid analytic variety.

Keywords

rigid analytic geometryp-adic Hodge theory

MSC classification

Primary: 14G22: Rigid analytic geometry
Type
Research Article
Information
Compositio Mathematica , Volume 154 , Issue 12 , December 2018 , pp. 2606 - 2642
Copyright
© The Author 2018 

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